2Vectors

IA Vectors and Matrices

2.3 Cauchy-Schwarz inequality

Theorem (Cauchy-Schwarz inequality). For all x, y ∈ R

n

,

|x · y| ≤ |x||y|.

Proof. Consider the expression |x −λy|

2

. We must have

|x − λy|

2

≥ 0

(x − λy) · (x − λy) ≥ 0

λ

2

|y|

2

− λ(2x · y) + |x|

2

≥ 0.

Viewing this as a quadratic in

λ

, we see that the quadratic is non-negative and

thus cannot have 2 real roots. Thus the discriminant ∆ ≤ 0. So

4(x · y)

2

≤ 4|y|

2

|x|

2

(x · y)

2

≤ |x|

2

|y|

2

|x · y| ≤ |x||y|.

Note that we proved this using the axioms of the scalar product. So this

result holds for all possible scalar products on any (real) vector space.

Example.

Let

x

= (

α, β, γ

) and

y

= (1

,

1

,

1). Then by the Cauchy-Schwarz

inequality, we have

α + β + γ ≤

√

3

p

α

2

+ β

2

+ γ

2

α

2

+ β

2

+ γ

2

≥ αβ + βγ + γα,

with equality if α = β = γ.

Corollary (Triangle inequality).

|x + y| ≤ |x| + |y|.

Proof.

|x + y|

2

= (x + y) · (x + y)

= |x|

2

+ 2x · y + |y|

2

≤ |x|

2

+ 2|x||y| + |y|

2

= (|x| + |y|)

2

.

So

|x + y| ≤ |x| + |y|.